Number system

\text{1. Find the value of} \, \, 98 \times 98

 

A. 9604

B. 9801

C. 9980

D. 9506

Ans: A

Explanation:

98 \times 98 = \left ( 98 \right )^2 = \left ( 100 - 2 \right )^2

 

\text{This is in the form of} \,\, \left ( a - b \right )^2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \left [ \left ( a - b \right )^2 = a^2 + b^2 - 2ab\right ]

 

\left ( 100 - 2 \right )^2 = \left ( 100 \right )^2 + \left ( 2 \right )^2 - 2 \times 100 \times 2

 

= 10000 + 4 - 400 = 9604

 

 

\text{2. Find the value of} \,\, 105 \times 105

 

A. 11025

B. 12045

C. 11990

D. 12590

Ans: A

Explanation:

105 \times 105 = \left ( 105 \right )^2 = \left ( 100 + 5 \right )^2

 

\text{This is in the form of} \, \left ( a + b \right )^2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \left [ \left ( a + b \right )^2 = a^2 + b^2 + 2ab\right ]

 

\left ( 100 + 5 \right )^2 = \left ( 100 \right )^2 + \left ( 5 \right )^2 + 2 \times 100 \times 5

 

= 10000 + 25 + 1000 = 11025

 

 

\text{3. Find the value of} \,\, 20 \times 19 + 20 \times 21

 

A. 600

B. 700

C. 800

D. 900

Ans: C

Explanation:

20 \times 19 + 20 \times 21 = 20\left ( 19 + 21 \right )

 

= 20(40)

 

= 800

 

 

4. Evaluate 500 – ? = 150 + 250

A. 50

B. 75

C. 90

D. 100

Ans: D

Explanation:

\text{Let 500} - x = 150 + 250

 

500 - x = 400

 

500 - 400 = x

 

x= 100

 

 

\text { 5. Total number of prime factors in the expression} \, \left ( 4 \right )^5 \times \left ( 9 \right )^6 \times \left \left ( 11 \right )^2 \, \text{is}

A. 20

B. 24

C. 25

D. 22

Ans: B

Explanation:

\left ( 4 \right )^5 \times \left ( 9 \right )^6 \times \left ( 11 \right )^2 = \left ( 2 \times 2 \right )^5 \times \left ( 3 \times 3 \right )^6 \times \left ( 11 \right )^2

 

= 2^5 \times 2^5 \times 3^6 \times 3^6 \times 11^2

 

= 2^{10} \times 3^{12} \times 11^2

 

\therefore \text{Total number of prime factors are } = 10 + 12 + 2 = 24

 

 

6. 9548 + 7314 = 8362 + ?

A. 8500

B. 8000

C. 6000

D. 9000

Ans: A

Explanation:

\text{Let} \,\, 9548 + 7314 = 8362 + x

 

16862 = 8362 + x

 

16862 - 8362 = x

 

8500 = x

 

 

7. The total number of even prime numbers are

A. 0

B. 1

C. 2

D. 3

Ans: B

Explanation:

\text{One and only even prime number is 2}

 

 

\text{8.} \,\, 31 \times 31 + 19 \times 19 + 2 \times 31 \times 19 = ?

 

A. 1500

B. 2000

C. 1800

D. 2500

Ans: D

Explanation:

31 \times 31 + 19 \times 19 + 2 \times 31 \times 19 = \left ( 31 \right )^2 + \left ( 19 \right )^2 + 2 \times 31 \times 19

 

= \left ( 31 + 19 \right )^2

 

= \left ( 50 \right )^2

 

= 2500

 

 

\text{9. If} \,\, x^2 - 190 = 6 \,\, \text{find the value of }x

 

A. 10

B. 12

C. 14

D. 16

Ans: C

Explanation:

x^2 - 190 = 6

 

x^2 = 6 + 190

 

x^2 = 196

 

x = \sqrt{196} = 14

 

 

10. Find the sum of first 10 numbers

A. 70

B. 55

C. 60

D. 95

Ans: B

Explanation:

\text{sum of the first n numbers = } \, \left [ 1 + 2 + 3 + .... + n = \frac{n\left ( n + 1 \right )}{2}\right ]

 

 

\text{sum of the first 10 numbers} = \frac{10\left ( 10 + 1 \right )}{2} = \frac{10\left ( 11 \right )}{2} = \frac{110}{2} = 55